Properties

Degree $2$
Conductor $6003$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 0.661·2-s − 1.56·4-s + 1.16·5-s + 4.79·7-s − 2.35·8-s + 0.768·10-s + 0.481·11-s − 3.88·13-s + 3.17·14-s + 1.56·16-s − 7.41·17-s + 4.27·19-s − 1.81·20-s + 0.318·22-s − 23-s − 3.65·25-s − 2.57·26-s − 7.49·28-s + 29-s − 0.692·31-s + 5.75·32-s − 4.90·34-s + 5.57·35-s − 5.71·37-s + 2.82·38-s − 2.73·40-s − 6.42·41-s + ⋯
L(s)  = 1  + 0.468·2-s − 0.780·4-s + 0.519·5-s + 1.81·7-s − 0.833·8-s + 0.243·10-s + 0.145·11-s − 1.07·13-s + 0.849·14-s + 0.390·16-s − 1.79·17-s + 0.979·19-s − 0.405·20-s + 0.0679·22-s − 0.208·23-s − 0.730·25-s − 0.504·26-s − 1.41·28-s + 0.185·29-s − 0.124·31-s + 1.01·32-s − 0.841·34-s + 0.942·35-s − 0.939·37-s + 0.458·38-s − 0.432·40-s − 1.00·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6003} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6003,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 - 0.661T + 2T^{2} \)
5 \( 1 - 1.16T + 5T^{2} \)
7 \( 1 - 4.79T + 7T^{2} \)
11 \( 1 - 0.481T + 11T^{2} \)
13 \( 1 + 3.88T + 13T^{2} \)
17 \( 1 + 7.41T + 17T^{2} \)
19 \( 1 - 4.27T + 19T^{2} \)
31 \( 1 + 0.692T + 31T^{2} \)
37 \( 1 + 5.71T + 37T^{2} \)
41 \( 1 + 6.42T + 41T^{2} \)
43 \( 1 + 8.76T + 43T^{2} \)
47 \( 1 + 0.616T + 47T^{2} \)
53 \( 1 + 6.31T + 53T^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 + 6.33T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + 6.45T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 2.44T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + 1.71T + 89T^{2} \)
97 \( 1 - 3.96T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.78340653686371494415913186511, −7.06103880628743883924309956717, −6.11478913398546533709123734517, −5.28424446167282234235491423319, −4.81130864098938350387439913618, −4.40294374394082541533989124835, −3.33347115706459539930146826451, −2.21249200135021101925451740005, −1.53681641000682454756543949449, 0, 1.53681641000682454756543949449, 2.21249200135021101925451740005, 3.33347115706459539930146826451, 4.40294374394082541533989124835, 4.81130864098938350387439913618, 5.28424446167282234235491423319, 6.11478913398546533709123734517, 7.06103880628743883924309956717, 7.78340653686371494415913186511

Graph of the $Z$-function along the critical line